SIR-SI model with a Gaussian transmission rate: Understanding the dynamics of dengue outbreaks in Lima, Peru

Introduction Dengue is transmitted by the Aedes aegypti mosquito as a vector, and a recent outbreak was reported in several districts of Lima, Peru. We conducted a modeling study to explain the transmission dynamics of dengue in three of these districts according to the demographics and climatology. Methodology We used the weekly distribution of dengue cases in the Comas, Lurigancho, and Puente Piedra districts, as well as the temperature data to investigate the transmission dynamics. We used maximum likelihood minimization and the human susceptible-infected-recovered and vector susceptible-infected (SIR-SI) model with a Gaussian function for the infectious rate to consider external non-modeled variables. Results/principal findings We found that the adjusted SIR-SI model with the Gaussian transmission rate (for modelling the exogenous variables) captured the behavior of the dengue outbreak in the selected districts. The model explained that the transmission behavior had a strong dependence on the weather, cultural, and demographic variables while other variables determined the start of the outbreak. Conclusion/significance The experimental results showed good agreement with the data and model results when a Bayesian-Gaussian transmission rate was employed. The effect of weather was also observed, and a strong qualitative relationship was obtained between the transmission rate and computed effective reproduction number Rt.

• The probability b h of infection from mosquito to human per bite is given by • The mortality rate µ v of an adult mosquito is given by S2 Appendix. Analysis of the model We analyzed the disease-free equilibrium point with exogenous and temperature variable terms. The basic reproduction number R 0 was computed by using the next-generation matrix evaluated at the disease-free equilibrium point [1][2][3]. At this equilibrium point, the vector population (i.e., mosquitoes) tends to decrease because of the environmental conditions, natural lifespan, and artificial external conditions such as insecticides [4] or control methods [4]. In addition, we considered the progression of cases by using the effective reproduction number R t .
Proof. First, (2)-(6) are reduced to Linearizing the above equations and evaluating the linear form at the disease-free where the matrix J is the Jacobian associated with (25) evaluated at E df . By simplicity, the matrix J is decomposed into J = F − V : The dynamics of (27) can be determined by the eigenvalues of the matrix J. The matrix M = F V −1 is of particular interest and is given by [2] The reproduction number R 0 is obtained from the spectral radius of M : , the following is obtained: Remark 1. Since we were interested in when R 0 is comparable to 1, we can use It is possible to observe that the transition is at R 0 = 1. The disease-free equilibrium point is asymptotically stable if and only if R 0 < 1, and the epidemic is settled if R 0 > 1.
In the case of Lima, the situation was strongly affected by external conditions including the environmental, inflow cases due to travelers (which were not considered in this model), and entomological conditions. An exogenous parameter was introduced to model external forces that affect the dynamics of the outbreak.
The epidemiological curves of the SIR-SI model (2) including the transmission rate (9) and exogenous parameter (11) were fitted to observed data by adjusting the parameters u, σ, and k (related to the β ex ). We used a maximum likelihood function as a cost function in the optimization solver. We used a normal distribution function for the probability under the assumption that the observation error had a normal distribution.

Corollary 1. If the functions and values for
Proof. The proof of this corollary is obtained directly from the definitions of b, b h , b v , and β ex .
The basic reproduction number (24) measures the epidemic potential of a pathogen, which in this case is dengue [2]. It is defined as the average number of new infections created by an infectious individual in an entirely susceptible population. In practice, the situation evolves, so the data are used to estimate the effective reproduction number R t . The quantitative changes in the effective reproduction number need to be monitored to determine when it approaches the critical threshold R t = 1 [5]. In the case of Lima, the reports of few cases would imply a small R 0 . However, as the outbreak evolves, R t can increase and even exceed the critical value of R t = 1.

S3 Appendix. Case progression for an infected case
Here, we consider the temperature-dependent SIR-SI model with an exogenous variable as expressed in (2). The total number of cases up to time t is represented by Q(t) and is calculated as follows: To consider the evolution of R t , we followed the approach of Bettencourt and Ribeiro [5]. We considered that epidemic reports generally state the occurrence of new infected cases within the period τ . Hence, the total number of cases is given by The change in cases between t and t + τ is obtained by where p = Iv I h . For theoretical purposes, we can consider the number of mosquitoes to remain constant from the time t to t + τ . Hence, I v (t + τ ) = I v (t). Thus, in (32) the value of p must satisfy Introducing (33) to (31) obtains where the effective reproduction number R t (t) can be calculated directly as The number of infectious cases decreases if R t < 1 or, equivalently, In general, the assumption that I v (t) remains constant during the evaluation period is adequate for emerging infectious diseases (i.e., few cases within a much larger , we used Bettencourt and population), which is the case for Lima. To compute R t Ribeiro's [5] algorithm.